The problem with twp linear branches

Piecewise fractional linear maps wzth three or more branches have been studied in several papers. For many Moebius maps the shape of the density of their invariant measurs can be written down exactly. However, if just two branches are linear, no explicit form is known. In this paper a partial solution is offered.

💡 Research Summary

The paper addresses a gap in the theory of piecewise fractional‑linear (Möbius) transformations on the unit interval. While maps with three or more branches have been extensively studied and, for many such maps, the invariant measure’s density can be written in closed form using the concept of a natural dual, the case where exactly two branches are linear and the third is non‑linear has resisted explicit description.

The author begins by fixing a specific partition of the interval, (0<p_{1}=1/3<p_{2}=2/3<1). A map (T: